Completing Ausoni's computation: an unconditional description of $V(1)_*K(KU_p)$

Ausoni's study of the algebraic K-theory of the connective complex K-theory spectrum $ku$ (2010) determined the mod-$(p,v_1)$ homotopy $V(1)_*K(ku_p)$ for $p\ge 5$, and gave a conditional description of $V(1)_*K(KU_p)$ for the periodic $p$-complete spectrum, dependent on two hypotheses: a Hesselholt-conjectured commutative diagram of localization sequences in THH/TC, and an algebra-structure statement for the relative term $THH(ku_p\mid KU_p)$. Later work of Blumberg–Mandell (2020) established the localization sequences, and the Lichtenbaum–Quillen hypothesis used elsewhere in Ausoni's paper is settled by the norm residue theorem; but the multiplicative structure of the relative term remains open, as do the proposed Hesselholt–Madsen-style relative $TR/TC(ku_p\mid KU_p)$ computation, the primes $2$ and $3$, and integral computations of $K(ku)$. This record states the surviving core — the unconditional determination of $V(1)_*K(KU_p)$ for $p\ge 5$ — as the problem; the remaining items are neighboring open threads described in the Background section.

Background

Algebraic K-theory $K(A)$, originally defined for rings, extends to structured ring spectra — $E_\infty$-ring spectra, which are ring objects in spectra whose multiplication is commutative and associative up to coherently parametrized homotopies. The prime examples studied first were the connective complex K-theory spectrum $ku$, with $ku_* = \mathbb{Z}[u]$, $|u| = 2$ ($u$ is the Bott element), and its periodic localization $KU = ku[u^{-1}]$, obtained by inverting $u$; throughout, $ku_p$ and $KU_p$ denote the $p$-completions of $ku$ and $KU$. The main computational tool is the cyclotomic trace, a multiplicative refinement of the classical trace map from K-theory to Hochschild homology: it goes from $K(A)$ to topological Hochschild homology $THH(A)$ — the homotopy-theoretic analogue of the Hochschild homology of a ring, defined for ring spectra — and on to topological cyclic homology $TC(A)$, built from the topological restriction homology groups $TR^n(A)$, i.e. the homotopy groups of the fixed points of $THH(A)$ under the natural actions of the cyclic groups $C_{p^n}$ of order $p^n$; for many $p$-complete ring spectra these invariants determine the mod-$p$ and finer homotopy type of $K(A)$. Here $V(1)$ denotes the Smith–Toda complex: the cofiber of a self-map $v_1$ of degree $2p-2$ (the first periodicity operator of stable homotopy theory) on the mod-$p$ Moore spectrum; it exists as a homotopy-associative ring spectrum for primes $p\ge 5$ (it does not exist in the needed form at $p = 2$ or $3$), and $V(1)_*X$ is the homotopy of $X\wedge V(1)$, i.e. the mod-$(p,v_1)$ homotopy of $X$. Ausoni and Rognes first computed $V(1)_*K(\ell_p)$ for the $p$-complete Adams summand $\ell_p$ of $ku_p$ (the connective summand with $\ell_{p*} = \mathbb{Z}_p[v_1]$, where $v_1 = u^{p-1}$), exhibiting chromatic redshift: the $v_1$-periodic ring spectrum $\ell_p$ has K-theory carrying a non-nilpotent class $b$ with $b^{p-1} = -v_2$, where $v_2$, of degree $2p^2-2$, detects the second layer of periodicity — height-one input yielding height-two behavior. Ausoni then computed $V(1)_*K(ku_p)$, and, following the classical localization sequence $K(\mathbb{Z})\to K(ku)\to K(KU)\to \Sigma K(\mathbb{Z})$, sought a description of $V(1)_*K(KU_p)$. Because the transfer in the K-theory localization sequence is hard to control directly, Ausoni (following Hesselholt) proposed to control it through corresponding localization sequences in THH and TC, of the shape

$THH(\mathbb{Z}_p)\to THH(ku_p)\to THH(ku_p\mid KU_p)\to \Sigma THH(\mathbb{Z}_p)$

and its TC analogue, involving a relative term $THH(ku_p\mid KU_p)$ analogous to the relative topological cyclic homology $TC(A\mid K)$ that Hesselholt and Madsen introduced for a discrete valuation ring $A$ with fraction field $K$ and computed in terms of de Rham–Witt cohomology (the $p$-adic, Frobenius-twisted analogue of the de Rham complex) with log poles. Ausoni's Theorem 8.3 states the expected $P(b)$-module structure of $V(1)_*K(KU_p)$ (with $P(b)$ a polynomial algebra on the higher Bott element $b$ of degree $2p+2$; following the source's conventions, $P(x_1,\dots)$ and $E(x_1,\dots)$ denote the polynomial and exterior $\mathbb{F}_p$-algebras on the listed generators, and $P_{p-1}(u)$ the truncated polynomial algebra $P(u)/(u^{p-1})$), but only under two hypotheses: that such a commutative diagram of localization sequences (his diagram (8.1)) exists, and that his additive identification $V(1)_*THH(ku_p\mid KU_p)\cong P_{p-1}(u)\otimes E(d,\lambda_1)\otimes P(\mu_1)$ is an isomorphism of algebras (his (8.2)); the latter is what produces the decisive relation $j_*(b)d = j_*(a_1)$ (here $j$ denotes the map induced by the localization $ku_p\to KU_p$, in K-theory and, via diagram (8.1), in THH; $a_1\in V(1)_{2p+3}K(ku_p)$ is one of the classes in Ausoni's Theorem 8.1 computation of $V(1)_*K(ku_p)$; and $d\in V(1)_1K(KU_p)$ is the class mapping to the unit $1\in V(1)_0K(\mathbb{Z}_p)$ under the connecting homomorphism of the localization sequence). Two neighboring open threads in the same paper frame the core: Propositions 1.3–1.4 assumed the Lichtenbaum–Quillen Conjecture for the homotopy fiber of $K(\mathbb{Z})\to K(\mathbb{Z}_p)$ (the prediction that its $V(1)$-homotopy is finite and concentrated below degree $2p-1$); and Ausoni remarked that an alternative route would be a direct relative computation of $TR^\bullet(ku_p\mid KU_p)$ and $TC(ku_p\mid KU_p)$ — an idea later given framework by the logarithmic $THH$/$TC$ of Rognes–Sagave–Schlichtkrull, which formalizes such relative terms through prelogarithmic structures on ring spectra — which he expected to be easier to handle than the absolute $TR^\bullet(ku_p)$ computation it avoids, while regarding integral or mod-$p$ (non-$V(1)$-localized) computations of $K(ku)$ as algebraically intractable with then-current techniques.

Problem Statement

Determine, unconditionally, the algebraic K-theory of the periodic complex K-theory spectrum, completing the program of Ausoni's 2010 paper: for primes $p\ge 5$, establish the expected description of $V(1)_*K(KU_p)$ as a $P(b)$-module by removing the remaining hypothesis of Ausoni's Theorem 8.3 — that is, prove that the identification of the relative term satisfies the multiplicative structure (the isomorphism $V(1)_*THH(ku_p\mid KU_p)\cong P_{p-1}(u)\otimes E(d,\lambda_1)\otimes P(\mu_1)$ of algebras for the relative term appearing in the now-proven Blumberg–Mandell localization sequences, and hence the relation $j_*(b)d = j_*(a_1)$ lifting to $V(1)_*K(KU_p)$), or otherwise determine $V(1)_*K(KU_p)$ without hypotheses.

Two neighboring directions recorded in the source are context for this problem, not requirements of it. Ausoni's Remark 8.4 speculated on the existence of a relative term $TC(ku_p\mid KU_p)$ fitting into a localization sequence $TC(\mathbb{Z}_p)\to TC(ku_p)\to TC(ku_p\mid KU_p)\to \Sigma TC(\mathbb{Z}_p)$, a Hesselholt–Madsen-style relative computation of $V(1)_*TR^n(ku_p\mid KU_p)$ and $V(1)_*TC(ku_p\mid KU_p)$ from which the K-theory could be deduced; the source expected such a relative computation to be easier to handle than the absolute computation it avoids, but it has not been carried out. The source also noted that its computations fail at the primes $p = 2$ and $3$, where the Smith–Toda complex $V(1)$ does not exist, and that integral or mod-$p$ (non-$V(1)$-localized) computations of $K(ku)$ seemed algebraically intractable with then-current techniques.

References

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